The Art of Mathematical Thinking: Exploring the Mind of a Mathematician

12 min readOct 7, 2023
Photo by Andy Prokh | Source:

The question of how mathematicians think is closely related to the question, “How does a musician compose?” Similarly, this question is asked to learn how the creative process works. Those who are interested in computer science or especially artificial intelligence can give the correct answer to this question. A real mathematician is not interested in finding the answer to this question, he’s just busy doing math.

Unfortunately, there is no clear way to answer the question of how a mathematician thinks. But we can approach this question as follows; if you watched any chess tournament, the game’s analysis is shared in detail at the end of the match. When you examine the analysis, you will see a breaking point in each game. Similarly, mathematicians also experience a breaking point while working on a problem before finding a solution.

Therefore, it is helpful to analyze a few mathematical proofs to answer our question and pinpoint the breaking points. For example, as we all know, Euclid’s theorem states that there are infinitely many prime numbers.

The proof of this theorem is more beautiful than the theorem itself.

So how did Euclid realize that prime numbers are infinite? Euclid’s approach to the solution is fascinating.

First, Euclid assumed that prime numbers are finite. He then constructed a set where he wrote all the prime numbers and called the elements of the set P = {p1, p2, p3,…, pr}. So, by that assumption, any number other than these should not be prime. Then Euclid multiplied all the elements of the set P and added 1 to the product. Then he got a new number and called this number N.

Besides this, Euclid has some excellent math knowledge he has been gifted with, such as the Fundamental Theorem of Arithmetic.

For example, he knows that if a number is not prime, it can be broken down into prime factors. Thus, when he tried to factorize the number N, since N isn’t a prime number, N must be divisible by at least one prime number. However, all the primes are here, and they cannot divide it because of the plus 1. For example, 2 and 3 make prime numbers, and 2 x 3 = 6. 2 and 3 can perfectly divide the…




Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better.